How Can You Distinguish Between Two Quantum States Using a Measurement Basis?

[Kreizhn]

Homework Statement

Given two quantum preparations
$\frac{1}{\sqrt{2}} \left( |0\rangle + | 1 \rangle \right)$
$\frac{1}{\sqrt{2}} \left( |0\rangle - | 1 \rangle \right)$
Give a measurement that will distinguish between these two preparations with high probability.

The Attempt at a Solution

I'm thinking that there might be some other measurement basis with which I can apply in order to get a high probability of determining which is which, but I can't think of it.

 

[tim_lou]

Do you know what 0 and 1 means? are they eigenstates of say angular momentum, Lz, states of harmonic oscillator? or something else?

If you don't know what 0 1 are (except different energy eigenstates), you will just have to arbitrarily construct a Hermitian operator whose eigenstates are the ones above and call that a measurement. I'm quite sure this is not what the question wants.

 

[Kreizhn]

The original question is how to differentiate between the following states in a 2-dimensional Hilbert space:

$$\frac{1}{\sqrt2} \left( | 0 \rangle + e^{3i\pi/4} | 1 \rangle \right)$$
$$\frac{1}{\sqrt2} \left( | 0 \rangle + e^{7i\pi/4} | 1 \rangle \right)$$

and the hint suggested that I use a $\pi/4$ shifter $| 0 \rangle\langle 0 | + e^{i\pi/4} |1 \rangle \langle 1|$.

 

[turin]

Hint: is a projection operator Hermitian?